Final answer:
The expression ((4a + 18))(a) yields an integer value for any integer value substituted for 'a'. Therefore, there are infinitely many possible values that 'a' can take, as any integer value will satisfy the condition.
Step-by-step explanation:
The expression ((4a + 18))(a) represents a product of a and a polynomial in a. If this expression is an integer, that means a can be any integer that, when substituted into the expression, results in an integer value. The number 18 is divisible by 1, 2, 3, 6, 9, and 18. When looking for possible values for a, we must consider how these factors pair with potential a values to maintain the product as an integer.
For example, if a is 1, the expression becomes 4*1 + 18 which is equal to 22, which is an integer. However, the expression does not limit a to positive integers only; a could be negative or zero as well. Therefore, there are infinitely many integers that could be substituted for a to make the expression an integer, since any integer multiplied by an integer (including the resulting expression from the polynomial) is still an integer.